1. Field of the Invention
The present invention relates to guidance systems for surface-to-air missiles, and particularly to a method of generating an integrated, fuzzy-based guidance law for aerodynamic missiles that uses a strength Pareto evolutionary algorithm (SPEA) based approach and a Tabu search to determine the initial feasible solution for the algorithm to select between one of three fuzzy controllers implementing different guidance laws to issue a guidance command to the missile.
2. Description of the Related Art
Guidance technology of missiles includes many well known guidance laws which are regularly utilized. The guidance and control laws typically used in current tactical missiles are mainly based on classical control design techniques. These conventional control approaches, however, are often not sufficient to obtain accurate tracking and interception of a missile. Therefore, advanced control theory must be applied to a missile guidance and control system in order to improve its performance. Fuzzy control has suitable properties to eliminate such difficulties, however, at the present time, there is very limited research related to fuzzy missile guidance design.
Fuzzy logic has been applied to change the gain of the proportional navigation guidance (PNG) law. Such a fuzzy-based controller was also used in the design of guidance laws where the line of sight (LOS) angle and change of LOS angle rate are used as input linguistic variables, and the lateral acceleration command can be used as the output linguistic variable for the fuzzy guidance scheme. It is known that these fuzzy guidance schemes perform better than traditional proportional navigation or augmented proportional navigation schemes; i.e., these methods result in smaller miss distances and lower acceleration commands.
In the above, though, the parameters of the fuzzy guidance law are generated by trial and error, which consumes time and effort, as well as computational power, and the results are not necessarily optimal. Moreover, such conventional methods and systems use only one type of guidance through the entire interception range. Each of the classical guidance laws has a particular region of operation in which they are found to be superior to other guidance laws.
In general, the multi-objective missile guidance law design problem can be converted to a single objective problem through the linear combination of different objectives as a weighted sum. The important aspect of this weighted sum method is that a set of non-inferior (or Pareto-optimal) solutions can be obtained by varying the weights. Unfortunately, this requires multiple runs, with the number runs being equivalent to the number of desired Pareto-optimal solutions. Furthermore, this method cannot be used to find Pareto-optimal solutions in problems having a non-convex Pareto-optimal front.
Evolutionary algorithms, however, may be used to efficiently eliminate most of the difficulties of classical methods. Since they use a population of solutions in their search, multiple Pareto-optimal solutions can be found in one single run. A multi-objective evolutionary algorithm (MOEA) must be started with a feasible solution, which is usually obtained by trial and error, thus requiring very high computational time.
It would be desirable to make such a methodology more efficient through the usage of a systematic technique to get the initial feasible solution, such as through the usage of a Tabu search (TS). TS is a higher level heuristic algorithm for solving combinatorial optimization problems. It is an iterative improvement procedure which starts from any initial solution and attempts to determine a better solution. TS has recently become a well-established optimization approach that is rapidly spreading to a variety of fields.
Now referring to actual missile guidance and control, we assume for the sake of simplicity that a missile's motion is constrained in the vertical plane. Furthermore, the missile may be modeled as a point mass with aerodynamic forces applied at the center of gravity. Thus, from the missile's balanced forces shown in FIG. 2, the equations of motion for the missile can be written as:
                                          γ            .                    m                =                                            (                              L                +                                  T                  ⁢                                                                          ⁢                  sin                  ⁢                                                                          ⁢                  α                                            )                                      m              ⁢                                                          ⁢                              V                m                                              -                                    g              ⁢                                                          ⁢              cos              ⁢                                                          ⁢                              γ                m                                                    V              m                                                          (                  1          ⁢          a                )                                                      V            .                    m                =                                            (                                                T                  ⁢                                                                          ⁢                  cos                  ⁢                                                                          ⁢                  α                                -                D                            )                        m                    -                      g            ⁢                                                  ⁢            sin            ⁢                                                  ⁢                          γ              m                                                          (                  1          ⁢          b                )                                                      x            .                    m                =                              V            m                    ⁢          cos          ⁢                                          ⁢                      γ            m                                              (                  1          ⁢          c                )                                                      h            .                    m                =                              V            m                    ⁢          sin          ⁢                                          ⁢                      γ            m                                              (                  1          ⁢          d                )                                L        =                              1            2                    ⁢          ρ          ⁢                                          ⁢                      V            m            2                    ⁢                      S            ref                    ⁢                      C            L                                              (                  1          ⁢          e                )                                          C          L                =                              C                          L              ⁢                                                          ⁢              α                                ⁡                      (                          α              -                              α                o                                      )                                              (                  1          ⁢          f                )                                D        =                              1            2                    ⁢          ρ          ⁢                                          ⁢                      V            m            2                    ⁢                      S            ref                    ⁢                      C            D                                              (                  1          ⁢          g                )                                          C          D                =                              C            Do                    +                      k            ⁢                                                  ⁢                          C              L              2                                                          (                  1          ⁢          h                )            where L, D, and T represent the lift, drag and thrust forces acting on the missile, respectively, ρ is the air density, Sref is the reference surface area, Ym represents the missile heading angle, α represents the missile angle of attack, m represents the missile mass, Vm represents the missile velocity, g is the gravitational acceleration, xm and hm are the horizontal and vertical positions of the missile, respectively, CL represents the lift coefficient, and CD represents the drag coefficient.
The aerodynamic derivatives CLα, CD0 and k are given as functions of the Mach number M, while the thrust and the mass are functions of time. The angle of attack α is used as the control variable and the missile normal acceleration can be determined from:
                              a          m                =                                                            γ                .                            m                        ⁢                          V              m                                =                                                    (                                  L                  +                                      T                    ⁢                                                                                  ⁢                    sin                    ⁢                                                                                  ⁢                    α                                                  )                            m                        -                          g              ⁢                                                          ⁢              cos              ⁢                                                          ⁢                              γ                m                                                                        (        2        )            where the target is assumed to be a point mass with a constant velocity Vt and acceleration at. The direction and position of the target in the horizontal and vertical directions are determined from the following relations:
                                          γ            .                    t                =                              a            t                                V            t                                              (                  3          ⁢          a                )                                                      x            .                    t                =                              V            t                    ⁢          cos          ⁢                                          ⁢                      γ            t                                              (                  3          ⁢          b                )                                                      h            .                    t                =                              V            t                    ⁢          sin          ⁢                                          ⁢                                    γ              t                        .                                              (                  3          ⁢          c                )            
From the interception geometry shown in FIG. 3, the line of sight angle rate and the derivative of the relative distance between the missile and the target can be written as:{dot over (θ)}=(Vm sin(θ−γm)−Vt sin(θ−γt))/r  (4a){dot over (r)}=−Vm cos(θ−γm)+Vt cos(θ−γt).  (4b)
In the above, θ represents the line of sight angle, r represents the distance between the missile and the target, and Vt represents the target velocity. For any surface-to-air missile, there are three guidance phases. The first phase of the trajectory is called the “launch” or “boost” phase, which occurs for a relatively short time. The function of the launch phase is to take the missile away from the launcher base. At the completion of this phase, midcourse guidance is initiated. The function of the midcourse guidance phase is to bring the missile near to the target in a short time. The last few seconds of the engagement constitute the terminal guidance phase, which is the most crucial phase, since its success or failure determines the success or failure of the entire mission.
There are two basic guidance laws governing homing missiles: Pursuit Guidance (PG) and the Proportional Navigation Guidance (PNG). PG guides the missile to the current position of the target, whereas PNG orientates the missile to an estimated interception point. Therefore, PNG has smaller interception time than PG, but this method may show unstable behavior for excessive values of the navigation constant. Thus, it is recommended to use PNG in the launching phase in order to get the fastest heading to the target, since stability is not a relatively large problem in this stage, while using PG in the terminal phase.
Since PNG is used during the boost phase to direct the missile velocity to the predicted interception location, the missile velocity should be aligned with the predicted interception velocity. Therefore, the missile command should be a function of a velocity error angle σ and its derivative. In the terminal phase, the position error dominates the final miss distance, thus it is recommended to use PG. The missile command must thusly be a function of the heading error δ in order to have a stable system with a minimum miss distance. During the midcourse phase, it is hoped that the missile reaches the terminal phase with the highest speed for the greatest distance possible and, at the same time, with a minimal heading error. Thus, the missile acceleration is a function of both variables.
The estimated value of the angle of this direction γp, can be obtained directly from the interception geometry in FIG. 3 as:
                              γ          p                =                  θ          -                                    atan              (                                                                    V                    T                                    ⁢                                      t                    p                                    ⁢                                      sin                    ⁡                                          (                      ϕ                      )                                                                                        r                  +                                                            V                      T                                        ⁢                                          t                      p                                        ⁢                                          cos                      ⁡                                              (                        ϕ                        )                                                                                                        ⁢                                                          )                        .                                              (        5        )            
The derivative of this angle is:
                                          γ            .                    p                =                              θ            .                    -                                                    V                T                            ⁢                                                t                  p                                ⁡                                  [                                                                                    -                                                  r                          .                                                                    ⁢                                                                                          ⁢                      sin                      ⁢                                                                                          ⁢                      φ                                        +                                                                  φ                        .                                            ⁡                                              (                                                                                                            V                              T                                                        ⁢                                                          t                              p                                                                                +                                                      r                            ⁢                                                                                                                  ⁢                            cos                            ⁢                                                                                                                  ⁢                            φ                                                                          )                                                                              ]                                                                                                      (                                                            V                      T                                        ⁢                                          t                      p                                                        )                                2                            +                              2                ⁢                r                ⁢                                                                  ⁢                                  V                  T                                ⁢                                  t                  p                                ⁢                cos                ⁢                                                                  ⁢                φ                            +                              r                2                                                                        (        6        )            where tp is the predicted time to intercept the target, which can be simply estimated as:
                              t          p                ≈                  -                                    r                              r                .                                      .                                              (        7        )            
As will be discussed in greater detail below, a is a variable defining the distribution of the membership function. It would be desirable to use a multi-objective evolutionary algorithm (MOEA) in order to generate missile guidance laws without requiring start points found via trial and error. Thus, a method of generating an integrated guidance law for aerodynamic missiles solving the aforementioned problems is desired.